What is the Rosary?
   Joyful Mysteries
   Luminous  Mysteries
   Sorrowful Mysteries
   Glorious Mysteries

The Apostle's Creed
Our Father
Hail Mary
Glory Be
O My Jesus (Fatima Prayer)

What is the Rosary?

Praying the Rosary is very simple. At first it will probably seem complicated, but after you pray it a couple of times, it becomes like second nature. In fact, it's much more complicated to talk about it than to pray it.
The word Rosary has two main meanings: the physical object, comprised of a string of beads, a medal, and a cross; and the prayer for which this physical object is used as a guide. You don't need the object to pray the Rosary, but the beads will help you keep track of where you are as you pray.  Moreover, the Rosary, as a physical object, is a sacred object, the presence of which in your home or pocket is, in a certain sense, a constant prayer offered to God, through Mary.
When we pray the Rosary, we start with the cross, say a prayer, then move on to the first bead and say a prayer, then move to the second bead and say a prayer, and continue in this manner until all of the beads are covered. (Note: You will find all of the prayers at the end of this article.)
The basic configuration of the prayer is simple: the Rosary consists of a Cross, a medal, individual beads, and consecutive beads. For the cross, we pray the Apostles' Creed; for the individual beads, the Our Father, and for the consecutive beads, the Hail Mary. After all of the beads have been covered, it is customary to say some concluding prayers (see below), including petitions (if the petitions have not already been stated), which can be understood to correspond to the medal. (Otherwise, the medal has no corresponding prayers.)
The Rosary can be divided into three parts: the introduction, the five decades, and the conclusion. The introduction consists of the cross and the four beads following it, which together mean the Apostles' Creed, one Our Father, and three Hail Marys. The five decades each consist of one individual bead and ten consecutive beads -- that is of one Our Father, and ten Hail Marys.
Each decade is devoted to a particular event in the life of Jesus or of the Virgin Mary. For the purposes of the Rosary, these are called mysteries. The focus is always on Jesus. But since the Virgin Mary, as the Mother of Jesus, had (and continues to have) a special relationship with Christ, the Rosary is designed to help us contemplate Christ and His love for humanity through His Mother -- and Our Mother -- on whom He has lavished so much love.
Since the Rosary has five decades, each of which corresponds to one mystery, there are five mysteries for each Rosary. Finally, there are four sets of five mysteries: 1) the Joyful Mysteries, 2) Luminous Mysteries, 3) the Sorrowful Mysteries, and 4) the Glorious Mysteries.
The mysteries are the following:
(The custom is to pray each set of mysteries on the days of the week indicated -- but that is just a custom.)

The Apostle's Creed

I believe in God, the Father Almighty, Creator of Heaven and earth. I believe in Jesus Christ, His only Son, our Lord. He was conceived by the power of the Holy Spirit and was born of the Virgin Mary. He suffered under Pontius Pilate, was crucified, died, and was buried. He descended to the dead. On the third day He rose again. He ascended into Heaven, and is seated at the right hand of the Father. He will come again to judge the living and the dead. I believe in the Holy Spirit, the Holy Catholic Church, the Communion of Saints, the forgiveness of sins, the resurrection of the body, and life everlasting. Amen.

Our Father

Our Father, Who are in Heaven; hallowed be Your name; Your kingdom come; Your will be done on earth as it is in Heaven. Give us this day our daily bread; and forgive us our trespasses as we forgive those who trespass against us, and lead us not into temptation; but deliver us from evil. Amen.

Hail Mary

Hail Mary, full of grace, the Lord is with you; blessed are you among women, and blessed is the fruit of your womb, Jesus. Holy Mary, Mother of God, pray for us sinners, now and at the hour of our death. Amen.

Glory Be:

Glory be to the Father, and to the Son, and to the Holy Spirit, as it was in the beginning, is now, and ever shall be, world without end. Amen.

O My Jesus (Fatima Prayer):

O my Jesus, forgive us our sins, save us from the fires of hell, and lead all souls to Heaven, especially those in most need of Your Mercy. Amen.
(Note: According to the testimony of the Fatima visionaries, Our Lady herself had requested in one of her apparitions that we pray the O My Jesus prayer at the end of each decade. The apparent universal willingness of the Catholic faithful around the world to comply with this wish is a living testimony to the authenticity of the visionaries' claim.)

Hail, Holy Queen:

Hail, Holy Queen, Mother of mercy, our life, our sweetness and our hope. To thee do we cry, poor banished children of Eve. To thee do we send up our sighs, mourning and weeping in this valley of tears. Turn then, most gracious advocate, thine eyes of mercy towards us. And after this our exile, show unto us the blessed Fruit of thy womb, Jesus. O clement, O loving, O sweet Virgin Mary.
- Pray for us, O holy Mother of God.
- That we may be made worthy of the promises of Christ

The Rosary Prayer:

Let Us Pray: O God, Whose only-begotten Son, by His life, death and resurrection, has purchased for us the rewards of eternal life, grant, we beseech You, that meditating upon these Mysteries of the Holy Rosary of the Blessed Virgin Mary, we may imitate what they contain, and obtain what they promise, through the same Christ, our Lord. Amen.

The Fifteen Promises of the Virgin Mary to those who recite the Rosary

(Revealed to St. Dominic and Blessed Alan):
1) Whoever shall faithfully serve me by the recitation of the Rosary, shall receive signal graces.
2) I promise my special protection and the greatest graces to all those who shall recite the Rosary.
3) The Rosary shall be a powerful armor against hell, it will destroy vice, decrease sin, and defeat heresies.
4) It will cause virtue and good works to flourish; it will obtain for souls the abundant mercy of God; it will withdraw the hearts of people from the love of the world and its vanities, and will lift them to the desire of eternal things. Oh, that souls would sanctify themselves by this means.
5) The soul which recommends itself to me by the recitation of the Rosary, shall not perish.
6) Whoever shall recite the Rosary devoutly, applying themselves to the consideration of its Sacred Mysteries shall never be conquered by misfortune. God will not chastise them in His justice, they shall not perish by an unprovided death; if they be just, they shall remain in the grace of God, and become worthy of eternal life.
7) Whoever shall have a true devotion for the Rosary shall not die without the Sacraments of the Church.
8) Those who are faithful to recite the Rosary shall have during their life and at their death the light of God and the plentitude of His graces; at the moment of death they shall participate in the merits of the Saints in Paradise.
9) I shall deliver from purgatory those who have been devoted to the Rosary.
10) The faithful children of the Rosary shall merit a high degree of glory in Heaven.
11) You shall obtain all you ask of me by the recitation of the Rosary.
12) All those who propagate the Holy Rosary shall be aided by me in their necessities.
13) I have obtained from my Divine Son that all the advocates of the Rosary shall have for intercessors the entire celestial court during their life and at the hour of death.
14) All who recite the Rosary are my children, and brothers and sisters of my only Son, Jesus Christ.
15) Devotion of my Rosary is a great sign of predestination.

"Say the Rosary every day, to obtain peace for the world." -- Our Lady of Fatima, 1917

Chaplet of Divine Mercy

The Chaplet of Divine Mercy is a prayer that Jesus gave to St. Faustina one morning in September 1935.  He told her - “This prayer will serve to appease my wrath.”

It is recited using a rosary.

Begin with:

The Our Father, the Hail Mary, and the Apostles Creed.

On the large beads before each decade, say:
Eternal Father, I offer You the Body, Blood, Soul and Divinity of Your dearly beloved Son, Our Lord Jesus Christ, in atonement for our sins and those of the whole world.

On the ten small beads of each decade, say:
For the sake of His sorrowful Passion, have mercy on us and on the whole world.

Conclude with:
Holy God, Holy Mighty One, Holy Immortal One, have mercy on us and on the whole world.  (say three times)

Hail, Holy Queen
The Rosary Prayer
The Fifteen Promises of the Virgin Mary to those who recite the Rosary
Some Reflections on the Spirit of Praying the Rosary
Further Reading - Rosary
Leonine
Chaplet of Divine Mercy

Links
How to Pray

The Joyful Mysteries:
(Monday and Thursday; and the Sundays from 1st Sunday of Advent until Lent.)
1. The Annunciation to Mary
2. The Visitation of Mary
3. The Birth of our Lord Jesus Christ
4. The Presentation of the Child Jesus in the Temple
5. The Finding of Our Lord in the Temple

The Luminous Mysteries:
(Thursdays throughout the entire year.)
1. The Baptism of Jesus in the River Jordan
2. The Wedding at Cana, Christ Manifested
3. The Proclamation of the Kingdom of God
4. The Transfiguration of Jesus
5. The Institution of the Holy Eucharist

The Sorrowful Mysteries:
(Tuesday and Friday; and the Sundays of Lent.)
1. The Agony of Christ in the Garden
2. The Scourging at the Pillar
3. The Crowning with Thorns
4. The Carrying of the Cross
5. The Crucifixion and Death of Our Lord on the Cross

Glorious Mysteries:
(Wednesday and Saturday; and the Sundays from Easter until Advent.)
1. The Resurrection of Our Lord
2. The Ascension of Our Lord
3. The Descent of the Holy Spirit upon the Apostles
4. The Assumption of the Blessed Virgin Mary into Heaven
5. The Coronation of Our Lady as Queen of Heaven and Earth

When we pray the Joyful Mysteries, the first decade corresponds to the Annunciation, the second decade to the Visitation, the third, to the Birth of Our Lord, the fourth, to the Presentation at the Temple, and the fifth, to the Finding in the Temple.
When we pray the Luminous Mysteries, the first decade corresponds to the Baptism of Jesus, the second, to the Wedding at Cana, the third, to the Proclamation of the Kingdom, the fourth, to the Transfiguration, and the fifth, to the Institution of the Eucharist.
When we pray the Sorrowful Mysteries, the first decade corresponds to the Agony of Christ in the Garden, the second, to Scourging of Christ, the third, to the Crowning with Thorns, the fourth, to the Carrying of the Cross, and the fifth, to the Crucifixion and Death of Our Lord.
When we pray the Glorious Mysteries, the first decade corresponds to the Resurrection of Our Lord, the second, to the Ascension of Our Lord, the third, to the Descent of the Holy Spirit upon the Apostles, the fourth, to the Assumption of the Virgin Mary, and the fifth, to the Coronation of the Virgin Mary.

Some Reflections on the Spirit of Praying the Rosary
The above is an explanation of the Rosary from a technical standpoint, but it is equally important to consider the spirit with which the Rosary should be prayed.
We should always keep in mind that the goal of the Rosary is Christ. When we pray the Rosary, we approach Christ through Mary -- His Mother and our Mother -- because of her special connection with Him and with us. It is beyond the scope of this article to explore all the reasons for devotion to Mary, but volumes could be written on this subject, as well as on the biblical foundations of Marian devotion.
Suffice it to say here that Mary is the Mother of Christ, and therefore is also our Mother, because we are a part of Christ. A family has to be very dysfunctional when the children do not want to honor their mother. In a very real sense, when we honor Mary, we do what the Fourth Commandment expects of us. Also, it was God Himself who kept Mary free from all stain of sin, made her His own Mother, raised her into Heaven after the completion of her earthly life, crowned her Queen of Heaven, and made her the mediator of grace between Christ and humanity. If God Himself has seen fit to honor Mary so abundantly, we do only what God Himself does when we too honor Mary. But we do not honor Mary for her own sake -- we honor her to glorify God, whose work she is.
Since Mary was always free from sin, her whole life was always centered on God. When we turn to her in prayer, she will immediately guide us to Christ, because she has never had a thought that was not an act of worship for God. Her whole being has always been defined by complete worship of God; she has never experienced another mode of existence. When we open our souls to her, she molds them to dispose them for the perfect love and worship of God.
After the human nature of Christ, Mary is the creature closest to God. Moreover, while Christ is the mediator of all grace between God and creation, Mary is the mediator of all grace between Christ and humanity. Consequently, Mary is a powerful intercessor for all who turn to her. God usually works through His creatures -- the Gospel is preached by people, Bibles are printed and distributed by people, churches are built by people. In the same way, many things will not happen unless people pray for them. That is why people need to pray for each other. If we believe that our prayers help each other in this life, why should we doubt that the prayers of those who have reached Heaven can also help us. If a person's prayer is efficacious while they are still in this world, why should their prayer not be even more efficacious when they are face to face with God, in eternal glory?
The history of the Church attests that the intercessions of Our Lady do not fail. When we pray to her, we can be sure that our prayers will be answered. Perhaps the answer will not come exactly in the form that we would like -- but in the end we will find ourselves more blest than ever, and we will find ourselves drawing closer and closer to God in worship and in love.
As we pray the individual prayers in the Rosary, we should say them slowly, either concentrating on the meaning of the words themselves, or meditating on the mystery of the decade. We should take great care not to allow praying the Rosary to degenerate into a mere habit, done fast, without much attention. Prayer is not magic. The words will do nothing for us if we do not pray them from the heart. A Rosary said hurriedly, with little attention, will have almost no benefit for us. But a Rosary said sincerely, with genuine devotion, will leave a lasting mark on our spiritual development.
If possible, we should meditate on each mystery either before, or after praying the decade, pausing for a minute or so, in order to reflect on meaning of the mystery.
Finally, whenever we pray the Rosary, we should make sure to state our petitions at some point, so that we know what we are praying for, and so that we can respond to Our Lady's desire to grant us more and more graces through the Rosary.

Further Reading
Many books have been written on the Blessed Virgin Mary. One of the best explorations of devotion to Our Lady through the Rosary is The Secret of the Rosary by St. Louis de Montfort (available through Tan Publishers). St. Louis de Montfort is known as the great Marian saint of the modern era. Though the style of his writings has not aged well, his message of Marian devotion has endured the test of time, and he continues to be one of the foremost authorities on devotion to the Virgin Mary. His book Devotion to the Blessed Virgin Mary is also one of the great spiritual classics of the Church.

Leonine Prayers
(Recited at the end of every mass for our men and women in the military.)

Salve Regina

Hail Holy Queen, Mother of mercy, our life, our sweetness, and our hope!  To thee do we cry, poor banished children of Eve; to thee do we send up our sighs, mourning and weeping in this valley of tears!  Turn, then, most gracious Advocate, thine eyes of mercy towards us; and after this, our exile, show unto us the blessed fruit of thy womb, Jesus.  O clement, O loving, O sweet Virgin Mary!

V. Pray for us, O holy Mother of God.
R. That we may be made worthy of the promises of Christ.

Let Us Pray

O God, our Refuge and our strength, look down with favor upon Thy people, who cry to Thee; and through the intercession of the glorious and Immaculate Virgin Mary, Mother of God, of her spouse, Blessed Joseph, of thy holy apostles, Peter and Paul, and of all the Saints, mercifully and graciously hear the prayers which we pour forth to Thee, for the conversion of sinners, and the liberty and exaltation of the Holy Mother Church.  Through Christ Our Lord.  Amen.

St. Michael the Archangel, defend us in battle; be our protection against the malice and snares of the devil.  We humbly beseech God to command him; and do thou, O Prince of the heavenly host, by the divine power thrust into hell Satan and the other evil spirits who roam through the world seeking the ruin of souls.  Amen.

Indulgence of 300 days - Pope Leo XIII.
Add the invocation: Most Sacred Heart of Jesus, have mercy on us. (three times)